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arXiv · 1211.6937

Self-commutators of Toeplitz operators and isoperimetric inequalities

Steven R. Bell·Timothy Ferguson·Erik Lundberg
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Abstract

For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.

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Steven R. Bell, Timothy Ferguson, Erik Lundberg. 2014-11-12. Self-commutators of Toeplitz operators and isoperimetric inequalities. https://doi.org/10.3318/pria.2014.114.03

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